Fisher Information and Jeffreys Prior for Bernoulli Trials

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Objective Bayes / Fisher Information.

Your Task
Jeffreys' non-informative prior is invariant under monotonic reparameterization and is defined by $p(\theta) \propto \sqrt{\mathcal{I}(\theta)}$, where $\mathcal{I}(\theta)$ is the Fisher Information.
(1) Compute the Fisher Information $\mathcal{I}(\theta)$ for the Bernoulli likelihood $f(x \mid \theta) = \theta^x (1-\theta)^{1-x}$.
(2) Derive Jeffreys' prior for $\theta$ and identify the corresponding Beta distribution parameters.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.